Among the multiple effects of Brexit, changes in migration and mobility across Europe were expected. Several studies have analysed these aspects, mostly from the point of view of perceptions, motivations, economic effects, scenarios, and changes in migration from Central and Eastern European countries. In this study we propose an analysis of migration and cross-border mobility using an integrated data-driven approach. We investigate official statistics from Eurostat, together with non-traditional data, to give a more complete view of the changes after Brexit, at EU and regional level. Specifically, we employ scientific publication and Crunchbase data to study highly-skilled migration, Twitter and Air Passenger data to investigate monthly trends. While main trends are preserved across datasets, with a general decrease in migration towards the UK immediately after the referendum approval, we are able to also observe more fine grained trends specific to some data or regions. Furthermore, we relate the changes in mobility observed from Air Passenger data with attention to Brexit from Google Trends data.